Maths - Splitting an equation understanding help

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BrandonS03
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So I got to a part in a question involving vectors where:
-bx/2 ax-kax kbx=-b/2 3a
Where a,b are vectors and x and k are scalar quantities where Im trying to find the value of k.
I read that you can split this equation into two simultaneous equations:
-bx/2 kbx=-b/2
and
ax-kax=3a
I know how to solve this by substitution however I dont understand how the equation can be split into the pair of simultaneous equations. How can the long equation be split into two? Any help in understanding why and how we can do this and maybe give me a few other examples of equations where this can happen. Thanks
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Benjamin213
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Well much like you can add two simultaneous equations together you do the opposite to split them. Imagine you have the simple simultaneous equations of 1) 3x+2y=36 and 2) 5x+4y=64 you can add the components together into one equation being (3x)+(5x)+(2y)+(4y)=100 or 8x+6y=100.

In the example you used, the two unlike vectors a and b have been split into simultaneous equations so you can get the vectors in the form of the scalar quantities.
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BrandonS03
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Any other explanations? I like to hear multiple explanations to fully understand why we can do this. Thanks
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mqb2766
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(Original post by BrandonS03)
Any other explanations? I like to hear multiple explanations to fully understand why we can do this. Thanks
When a and b are unit vectors
a = (1,0)
b = (0,1)
then, by definition, you have two split equations in the way you describe.
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